
for \(q \ge 3\) . Thus, it follows from Bernstein's inequality [1]} that for any \(a \ge 1\)
\(\begin{split}& \mathbb {P}\left(\left. \sum _{l = 0}^{L}{\overline{\varepsilon }^O_{\hat{\tau }^O_{L,l} :\hat{\tau }^O_{L,l + 1}} \varepsilon ^E_{\hat{\tau }^O_{L,l + 1}}} \ge a \log \left(K_{\max }\right) \overline{\sigma }\left(\sum _{l = 1}^{L} \left(\overline{\varepsilon }^O_{\hat{\tau }^O_{L,l} :\hat{\tau }^O_{L,l + 1}}\right)^2\right)^{1/2} (\sqrt{2} + c) \right|\mathcal {E}^O\right)\\\le & \exp (-a \log \left(K_{\max }\right)) = K_{\max }^{-a}.\end{split}\) 
